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Lesson 1 MA 15200
Sets of Numbers:
1. Natural Numbers: *{1, 2, 3, 4, ...}
*Note: This type of set notation is called roster set notation.
2. Whole Numbers: {0, 1, 2, 3, ...}
Natural numbers + Zero
3. Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
Whole numbers + Opposites of wholes (negatives)
a

4. Rational Numbers: *  | a is an integer and b is a nonzero integer 
b

*Note: This type of set notation is called set-builder notation.
Rational numbers include integers, fractions (proper, improper, or mixed
numbers), terminating decimals, and repeating decimals.
8
11
4
8  
 1.375
 0.8
1
8
5
2
13
  0.666... or  0.6
 1.181818.... or 1.18
3
11
5. Irrational Numbers: {x | x is a non-terminating or a non-repeating decimal}
Most irrational numbers are roots. Another well-known irrational number is  .
6. Real Numbers: {x | x is rational or irrational}
When every number from a first set of numbers (called set A) is also included in a second
set of numbers (called set B), then set A is called a subset of set B and is written A  B .
Example 1: True or False?
a)
If A  {2,5} and B  {1, 2,3, 4,5, 6} A  B
b)
If A = { all integers less than -10}, B = { all rational numbers greater
than -100}, A  B
a)
Natural numbers  Whole numbers
b)
Integers  Whole numbers
c)
Integers  Rational numbers
A prime number is a natural number greater than 1 divisible by only 1 and itself.
The first few prime numbers are 2, 3, 5, 7, 11, 13, 17.
1
A composite number is a natural number greater than 1 that is not prime.
The first few composite numbers are 4, 6, 8, 9, 10, 12.
An even number is an integer that is divisible by 2. An odd number is an integer than
is not divisible by 2.
Evens: {..., -4, -2, 0, 2, 4, ...}
Odds: {..., -3, -1, 1, 3, ...}
Example 2: True or False?
a) Even numbers  Whole numbers
b)
Odd numbers  Integers
c)
Prime numbers  Natural numbers
Relationship between Sets of Numbers and Examples
Real Numbers
5
 2, 3 ,  , 22, 3.8, 0,  2.4
9
Rational Numbers
7
11
,  12, 1, 0.98, ,  3.1
8
2
Non-integer Rational Numbers
12
1
 , 0.25, , 23.88,  0.45
7
3
Irrational Numbers
 ,  12 , 43 9
Integers
 4, 123, 9,  34
Negative Integers
Whole Numbers
 4,  99,  8,  51
0, 12, 40, 936
Zero
0
Natural Numbers
1, 13, 45, 60
2
Example 3: Given the following real numbers,
3
8
11
 , ,  2, 0,  2, , 3.4, 3,  2.83, 2 3,  6,  , 20
4
5
4
a)
Which numbers are whole numbers?
b)
Which numbers are integers?
c)
Which numbers are rational numbers?
d)
Which numbers are irrational numbers?
e)
Which numbers are even numbers?
f)
Which numbers are prime numbers?
Properties of Real Numbers:
If a, b, and c are real numbers,
1. Associative Properties for Addition and Multiplication
GROUPING
(a  b)  c  a  (b  c)
(ab)c  a(bc)
ORDER
2. Commutative Properties for Addition and Multiplication
ab  ba
ab  ba
3. Distributive Property of Multiplication over Addition SUM  PRODUCT
a(b  c)  ab  ac
a(b  c)  ab  ac
Note: This property applies if there are more than 2 terms within parentheses.
4. The Double Negative Rule
a
(  a )  a
a
1
Example 4: Determine which property justifies each statement.
a ) ab  3  3  ab
b)
( m  n ) 6  6( m  n )
c)
4(3  r )  12  4r
d)
5  ( x)  5  x
e)
4( xyz)  ( 4 x)( yz )
Graphing on a Number Line & Inequalities
Real Number Line:
-4
-2
0
2
4
6
Ex 1: Graph the prime numbers less than 10 on a number line.
0
2
4
6
8
3
Ex 2: Graph x  3
Note: A parenthesis is used in place of an open circle.
Ex 3: Graph x  2 Note: A bracket is used in place of a closed circle.
Ex 4: Graph x  0 or x  3
-4
-2
0
Ex 4: Graph this compound inequality, 
2
4
1
 x  4.3
2
Interval Notation is another way to represent an inequality or a graph of a portion of a
number line.
Unbounded intervals:
( a,  )
(
a
(, b]
Open, Half-open, and Closed intervals:
( a, b)
(
)
a
b
[ a, b)
[
a
[ a, b]
[
]
a
b
]
b
)
b
Ex 5: Write this inequality in interval notation and graph on a number line.
{x | x > 1}
Ex 6: Write the set of numbers represented on this number line as an inequality and in
interval notation.
(
]
3.5
5.7
4
Ex 7: Write the following as an inequality and graph on a number line. (,5]
The absolute value of a real number x ( |x| ) is the distance on a number line between 0
and the point with coordinate of x.
x  x when x  0
x   x when x  0
Ex 8: Write the following without an absolute value sign.
a )  12
b)
5
c)
  6 .5
d)
  5
e)
2  x if x  3
The distance between two coordinates a and b on a number line is defined as b  a .
Ex 9: Find the distance between the given coordinates.
a)  2 and 6
b)
 12 and  40
c)
12 and 444
5